Irrational Exponents! (For rather advanced mathematicians)?

I was thinking about the true nature of exponents last night, and was wondering what exponents really mean. Traditionally, we are told:

A^3 = A x A x A,

A^(3/2) = A^3 x A^(1/2) = A x A x A x root(A)

and so on....

For any rational exponent, this is just fine, as it can be expressed as a fraction, and then done as above.

We give this a meaning... something we can explain, namely:

Integer exponents mean: Multiply this number N amount of times

Roots, or fractional exponents mean: "Which number, multiplied N amount of times, gives you A?"

But for something like A^pi.... What does it MEAN?

We could perhaps say that if B = A^(pi) then ln(B) = (pi)ln(A), and look it up on a log table, or use a Taylor approximation to approximate the ln. For that matter, you could use a Taylor approximation on the original exponent... XD But that doesn't tell us what an irrational exponent really is. I mean, by definition an exponent is the amount of times a number is multiplied by itself. Then we introduced rational exponents, and said that the number at the bottom of the fraction was the root. This still leaves all the irrational numbers out of our understanding of exponents! I know they can be approximated to infinitesimal precision but still. I guess I'm looking for the true definition of A^b.... One that works and makes sense for all real and complex numbers.

Cheers,

M.G.